Term Rewriting System R:
[x, y, z]
f(+(x, 0)) -> f(x)
+(x, +(y, z)) -> +(+(x, y), z)

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(+(x, 0)) -> F(x)
+'(x, +(y, z)) -> +'(+(x, y), z)
+'(x, +(y, z)) -> +'(x, y)

Furthermore, R contains two SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering
       →DP Problem 2
         ↳AFS


Dependency Pair:

F(+(x, 0)) -> F(x)


Rules:


f(+(x, 0)) -> f(x)
+(x, +(y, z)) -> +(+(x, y), z)





The following dependency pair can be strictly oriented:

F(+(x, 0)) -> F(x)


There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(0)=  0  
  POL(F(x1))=  x1  
  POL(+(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem.
Used Argument Filtering System:
F(x1) -> F(x1)
+(x1, x2) -> +(x1, x2)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 3
             ↳Dependency Graph
       →DP Problem 2
         ↳AFS


Dependency Pair:


Rules:


f(+(x, 0)) -> f(x)
+(x, +(y, z)) -> +(+(x, y), z)





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳Argument Filtering and Ordering


Dependency Pairs:

+'(x, +(y, z)) -> +'(x, y)
+'(x, +(y, z)) -> +'(+(x, y), z)


Rules:


f(+(x, 0)) -> f(x)
+(x, +(y, z)) -> +(+(x, y), z)





The following dependency pair can be strictly oriented:

+'(x, +(y, z)) -> +'(x, y)


The following usable rule using the Ce-refinement can be oriented:

+(x, +(y, z)) -> +(+(x, y), z)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(+(x1, x2))=  1 + x1 + x2  
  POL(+'(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem.
Used Argument Filtering System:
+'(x1, x2) -> +'(x1, x2)
+(x1, x2) -> +(x1, x2)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
           →DP Problem 4
             ↳Argument Filtering and Ordering


Dependency Pair:

+'(x, +(y, z)) -> +'(+(x, y), z)


Rules:


f(+(x, 0)) -> f(x)
+(x, +(y, z)) -> +(+(x, y), z)





The following dependency pair can be strictly oriented:

+'(x, +(y, z)) -> +'(+(x, y), z)


The following usable rule using the Ce-refinement can be oriented:

+(x, +(y, z)) -> +(+(x, y), z)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(+(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem.
Used Argument Filtering System:
+'(x1, x2) -> x2
+(x1, x2) -> +(x1, x2)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
           →DP Problem 4
             ↳AFS
             ...
               →DP Problem 5
                 ↳Dependency Graph


Dependency Pair:


Rules:


f(+(x, 0)) -> f(x)
+(x, +(y, z)) -> +(+(x, y), z)





Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:00 minutes